I would like to install Wine and run a few simple Windows programs. Once upon a time I had it installed. Somewhere along the line, probably during a version upgrade, it vanished. Now when I use the Linux Mint 19.1 Software Manger, there is a rather confusing number of versions of Wine. I tried a few and nothing shows up on the Menu.

I tried running it from the command line and got:
art@art-Inspiron-5570:~$ sudo wine
[sudo] password for art:
it looks like wine32-development is missing, you should install it.
as root, please execute "apt-get install wine32-development"
Usage: wine PROGRAM [ARGUMENTS...] Run the specified program
wine --help Display this help and exit
wine --version Output version information and exit
art@art-Inspiron-5570:~$ sudo apt-get install wine32-development
Reading package lists... Done

And a bunch of lines later, and another attempt at running it got:
art@art-Inspiron-5570:~$ wine
Usage: wine PROGRAM [ARGUMENTS...] Run the specified program
wine --help Display this help and exit
wine --version Output version information and exit
art@art-Inspiron-5570:~$ wine --version
wine-3.6 (Ubuntu 3.6-1)
art@art-Inspiron-5570:~$ wine-3.6
wine-3.6: command not found

Okay, what we are dealing with here:
This is a Dell laptop Inspiron.
Intel Core i5 processsor (8th generation)
Running Linux Mint 19.1 with all of the updates.

Wee, that didn't work so well. Any ideas?

It looks like it's installed and working pretty well. As per the instructions you need to obtain your Windows programs and then invoke them using wine.

Try following this from Step 2: https://www.wikihow.com/Use-Wine-on-Linux

Maybe trying going to the actual directory where wine-3.6 resides (using the FIND command?) and try typing the following command to see if that works:

./wine-3.6

If it works then that probably means it's an issue with the PATH variable causing the system to NOT being able to find the program. You should be able to modify the PATH statement to append the 'Wine-3.6' directory to it. Some guru here will know exactly the correct syntax to do that.... something like echo $PATH, wine directory $PATH (but DON'T try that statement cause it's only a guess without me looking it up) :-)

I tried that site and followed the directions for configuring it, and got the dialogue box so selected Windows 10, clicked apply and then clicked on Okay, and as indicated, the dialogue box closed. However, the rest of the directions about starting a Windows program with Wine dont't work - as it right clicking on the .exe file and opening with Wine Program Loader - simply isn't there.

I have not been able to find a Wine-3.6. The closest one was Wine64, but trying to execute Wine64 resulted in ye olde Command Not Found response, which is the same response as trying to execute Wine-3.6.

OK.

Can you run the following command in a terminal:

There could be a handful of reasons why the "Wine Program Loader" isn't in the context menu. Let's worry about that once you've got your software running.

I tried that, and got this:
art@art-Inspiron-5570:~$ wine /path/to/wherever/your-program.exe
wine: cannot find '/path/to/wherever/your-program.exe'
art@art-Inspiron-5570:~$

I also removed that which I had installed and went to the Wine Web page and installed these lines as per instructions there.
This is Linux Mint, and is based on Ubuntu, so I used the Ubuntu version.

sudo apt-add-repository 'deb https://dl.winehq.org/wine-builds/ubuntu/ disco main'
sudo apt-add-repository 'deb https://dl.winehq.org/wine-builds/ubuntu/ bionic main'
(the above was for linux Mint 19.X)
sudo apt install --install-recommends winehq-stable
(the above was for the stable branch)

Trying to execute Wine-4.0 resulted in wine-4.0: command not found

You were suppose to replace '/path/to/wherever/your-program.exe' with the real location of your executable, the .exe that you were trying to right click on. I'm sorry if that wasn't clear.